4. Mp Se Structure Keiko bakes two sarne-sized pies to sell at a bake sale. She sells \( \frac{4}{6} \) of the blueberry pie. She sells \( \frac{2}{3} \) of the apple pie. Are the two fractions equivalent? Draw a visual model to support your answ

Yes, the two fractions \( \frac{4}{6} \) and \( \frac{2}{3} \) are **equivalent**. This means they represent the same portion of a whole, even though they may look different at first glance. ### **Understanding Equivalent Fractions** To determine if two fractions are equivalent, you can simplify them to their lowest terms or find a common denominator. 1. **Simplifying \( \frac{4}{6} \):** \[ \frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3} \] So, \( \frac{4}{6} \) simplifies to \( \frac{2}{3} \). 2. **Comparing with \( \frac{2}{3} \):** Since \( \frac{4}{6} \) simplifies to \( \frac{2}{3} \), both fractions are equivalent. ### **Visual Model** Let's use a visual representation with pies divided into equal parts to illustrate this: #### **Blueberry Pie (\( \frac{4}{6} \) sold)** Imagine the blueberry pie is divided into **6 equal slices**: ``` [ ][ ][ ][ ][ ][ ] ``` Out of these 6 slices, Keiko sells 4: ``` [S][S][S][S][ ][ ] ``` - **S** = Sold slice - **Empty** = Unsold slice #### **Apple Pie (\( \frac{2}{3} \) sold)** Now, imagine the apple pie is divided into **3 equal slices**: ``` [ ][ ][ ] ``` Out of these 3 slices, Keiko sells 2: ``` [S][S][ ] ``` #### **Comparing Both Pies** To easily compare the two fractions, let's adjust the apple pie to also have **6 slices** by multiplying both the numerator and the denominator by 2 (since \( \frac{2}{3} \times \frac{2}{2} = \frac{4}{6} \)): ``` [ ][ ][ ][ ][ ][ ] ``` Keiko sells 4 out of these 6 slices: ``` [S][S][S][S][ ][ ] ``` Now, both pies visually show that Keiko sold 4 out of 6 slices, confirming that \( \frac{4}{6} \) and \( \frac{2}{3} \) are indeed equivalent. ### **Conclusion** Both fractions represent the same portion of their respective pies: - \( \frac{4}{6} = \frac{2}{3} \) Therefore, the amounts sold from both the blueberry and apple pies are equivalent.